J Comb Optim (2014) 28:58–81DOI 10.1007/s10878-014-9734-0

The unconstrained binary quadratic programmingproblem: a survey

Gary Kochenberger · Jin-Kao Hao ·Fred Glover · Mark Lewis · Zhipeng Lü ·Haibo Wang · Yang Wang

Published online: 18 April 2014© Springer Science+Business Media New York 2014

Abstract In recent years the unconstrained binary quadratic program (UBQP) hasgrown in importance in the field of combinatorial optimization due to its applicationpotential and its computational challenge. Research on UBQP has generated a widerange of solution techniques for this basic model that encompasses a rich collection ofproblem types. In this paper we survey the literature on this important model, providingan overview of the applications and solution methods.

G. Kochenberger (B)School of Business Administration, University of Colorado at Denver, Denver, CO 80217, USAe-mail: gary.kochenberger@ucdenver.edu

J.-K. HaoLERIA, Université d’Angers, 2 Boulevard Lavoisier, 49045 Angers, Francee-mail: hao@info.iniv-angers.fr

F. GloverOptTek Inc., Boulder, CO 80302, USAe-mail: glover@opttek.com

M. LewisCraig School of Business, Missouri Western State University, Saint Joseph, MO 64507, USAe-mail: mlewis14@missouriwestern.edu

Z. LüSchool of Computer Science and Technology, Huazhong University of Science and Technology,Wuhan 430074, Chinae-mail: Zhipeng.lui@gmail.com

H. WangSanchez School of Business, Texas A&M International University, Laredo, TX 78041, USAe-mail: hwang@tamiu.edu

Y. WangSchool of Management, Northwestern Polytechnical University, 127 Youyi West Road,Xi’an 710072, Chinae-mail: sparkle.wy@gmail.com

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Keywords Unconstrained binary quadratic programs · Combinatorial optimization ·Metaheuristics

1 Introduction

The unconstrained binary quadratic programming (UBQP) problem is defined by

min xt Qxs.t. x ∈ S

where S represents the binary discrete set {0, 1}n or {−1, 1}n and Q is an n-by-n square, symmetric matrix of coefficients. This simple model is notable forembracing a remarkable range of applications in combinatorial optimization. Forexample, the use of this model for representing and solving optimization prob-lems on graphs, facility locations problems, resources allocation problems, clus-tering problems, set partitioning problems, various forms of assignment problems,sequencing and ordering problems, and many others have been reported in theliterature.

Even more remarkable is the fact that, once given a UBQP formulation, theseproblems can be solved by a UBQP method which is not specialized to exploit theproblem domain of any individual class of problems, to yield solutions whose qualityin many cases rivals or even surpasses the quality of the solutions produced by thebest specialized methods, while achieving this outcome with an efficiency that likewiserivals or surpasses the efficiency of leading specialized methods.

In this paper we survey the literature on UBQP, both its applications and solu-tion methods. While many important constrained nonlinear binary models have beenreported in the literature over the years, we focus our attention here on modelsthat naturally occur in the form of an unconstrained quadratic binary model andthose that have been re-cast into the form of UBQP. The paper is organized asfollows. In Sect. 2 we survey the range of applications that have been reportedin the literature. Section 3 then presents a survey of the solution methodologiesreported in the literature for solving UBQP. Section 4 highlights key theoretical workand this is followed by Sect. 5 which wraps up the paper with our summary andconclusions.

2 Applications

Some reported applications appear naturally in the form of UBQP while oth-ers are “re-cast” into the UBQP form by employing various transformations. Inthe sections below we examine these different categories of applications in turn.Within sections, we present applications in the chronological order in which theyappeared in the literature to give the reader a sense of when certain topics wereaddressed appeared in print, as well as progress made and trends in solutionmethodologies.

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2.1 Natural UBQP problems/applications

The literature on UBQP goes back to the 1960s where the topics of pseudo-Booleanfunctions and binary quadratic optimization were introduced by Hammer and Rudeanu(1968).

Early papers related to UBQP concern applications in finance (Laughhunn 1970),project selection (Rhys 1970), cluster analysis (Rao 1971), economic analysis (Ham-mer and Shlifer 1971), traffic management (Witzgall 1975) and computer aided design(Krarup and Pruzan 1978). While these applications actually take the form of con-strained quadratic binary programs, they are mentioned here due to their historicalrole in fostering an interest in quadratic binary applications and also because severalallow special cases that are precisely in the form of UBQP.

More recently many interesting applications that are expressed naturally in the formof UBQP have appeared in various papers. Barahona et al. (1988) formulate and solvethe problem of finding ground states of spin glasses with exterior magnetic fields, animportant problem in physics, as an instance of UBQP. Computational results revealthat the model produces high quality solutions to spin glass problems of realistic sizein reasonable amounts of computation time using 1980s technology.

Hansen and Jaumard (1990), in their work on the satisfiability problem, report theirexperience using the UBQP model as an approach for representing and solving small tomedium sized Max 2-sat problems. Computational studies validated the attractivenessof this approach to the Max 2-sat problem in terms of quickly producing high qualitysolutions.

Boros and Hammer (1991) discuss the use of UBQP as an approach for modelingthe Max-Cut problem. Their paper highlights the relationship between UBQP, Max-cut, Max 2-sat, and the Weighted Signed Graph Problem. The authors also presenta discussion of valid inequalities and facets for polyhedra that provide the basis forfurther computational and theoretical work.

Alidaee et al. (1994) discuss two machine scheduling problems in the contextof UBQP: (1) scheduling n jobs on a single machine to minimize total weightedearliness and tardiness, and (2) scheduling n jobs on two parallel identical processorsto minimize weighted mean flow time. In each case, the authors show how the problemscan be modeled in a straight-forward manner as an instance of UBQP.

Pardalos and Xue (1994) indicate how the maximum clique problem can be mod-eled as an instance of UBQP. The authors also discuss the relationship between themaximum clique problem, the maximum independent set problem, and the vertexcover problem, indicating how each can be represented by UBQP. Finally, the authorsprovide a survey of solution methods for the maximum clique problem.

De Simone et al. (1995), as in the earlier 1988 paper by Barahona, Grotschell,Junger and Reinelt, adopt the UBQP model as a representation for the problem offinding ground states for the spin glass problem. In this 1995 paper the authors use theUBQP model to compute exact ground states for Ising spin glasses on 2-dimensionalgrids with periodic boundary interactions, Gaussian bond distributions, and an exteriormagnetic field. Preliminary experiments with a branch and cut algorithm for optimizingthe UBQP form of the problem proved very promising, quickly producing high qualitysolutions to large spin glass instances.

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Bomze et al. (1999) discuss the maximum clique (MC) problem and how it canhow it can be modeled in a variety of ways including a representation in terms ofUBQP. The authors provide a very broad and in depth discussion of a variety ofapplications and of both exact and heuristic solution methods for the MC problem.Computational experience with various solution approaches to the MC problem is alsopresented.

Iasemidis et al. (2001) discuss the use of the UBQP model as part of a processemployed to predict the arrival of epileptic seizures. The entrainment between twobrain sites can be quantified from measures of electrical activity (EEG) of the brain.The UBQP model was successful in identifying the most entrained sites leading to theoptimal location of electrode sites. In clinical trials this procedure was successful inpredicting epileptic seizures 20–40 min in advance of their occurrence.

Alidaee et al. (2005) discuss the number partitioning problem where the objectiveis to assign numbers to subsets such that the sums of the numbers in each subset areas close as possible to one another. The authors show that the n = 2 subset case canbe modeled as an instance of UBQP and that problems with n > 2 can be modeled asa constrained version of UBQP. Extensions of the basic model along with computa-tional experience for the n = 2 case are presented indicating the attractiveness of theapproach.

Kochenberger et al. (2005a) discuss their experience with adopting the UBQPmodel to represent and solve Max 2-sat problems. Expanding the computational scopereported earlier by Hansen and Jaumard (1990) on UBQP and the Max 2-sat problem,they offer extensive computational experience on very large test problems with up to1,000 variables and more than 10,000 clauses. Employing a basic form of tabu searchto solve the UBQP instances, best known solutions to most test problems were foundin a few seconds of computation time.

Neven et al. (2008) discuss the use of quantum adiabatic algorithms, which rep-resent new approaches to NP-hard combinatorial problems, for solving the imagerecognition problem. The authors indicate how the pattern recognition problemof deciding whether two images contain the same object can be modeled as aninstance of UBQP, which they show is the general input format required by D-Wave superconducting quantum AQC processors. Computational experience was notreported.

Mahdavi Pajouh et al. (2013) discuss the use of the UBQP model for representingthe maximal independent set problem. The authors present an analysis of local max-ima properties along with relations between continuous local maxima of the quadraticformulation and the binary local maxima in the Hamming distance 1 and 2 neighbor-hoods. These results are then used to construct effective local search algorithms forthe maximum independent set problem.

Kochenberger et al. (2013) discuss the Max Cut problem and how the UBQP modelcan be effectively used to model and solve large scale instances. Using a tabu searchalgorithm, extensive computational testing is reported on problems with up to 10,000variables. Comparisons with other solution methods from the literature for the maxcut problem are provided, indicating the attractiveness of the UBQP/Tabu Searchapproach.

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Table 1 Illustrative KnownPenalties

Classical constraint Equivalent penalty

x + y ≤ 1 P(xy)

x + y ≥ 1 P(1 − x − y + xy)

x + y = 1 P(1 − x − y + 2xy)

x ≤ y P(x − xy)

x1 + x2 + x3 ≤ 1 P(x1x2 + x1x3 + x2x3)

2.2 UBQP via reformulation

The applications of the previous section illustrate the widespread usefulness of theUBQP model. The actual applicability of UBQP, however, is greatly extended dueto re-formulation procedures that re-cast a constrained problem into an equivalentunconstrained binary quadratic model. Many re-formulations are accomplished byincluding quadratic infeasibility penalties in the objective function as an alternative toexplicitly imposing constraints. In this manner a constrained model can be re-cast intothe form of UBQP. In fact, any linear or quadratic problem with linear constraints andbounded integer variables can in principle be re-formulated as UBQP using quadraticpenalties.

For several simple constraints, appropriate quadratic penalties are known in advanceand can be used straight away. Examples of such penalties are given in Table 1 whereP is a large positive scalar.

Note that the penalty term in each case is zero if the associated constraint is satisfied,and otherwise the penalty is positive. These penalties, then, can be directly employedas an alternative to explicitly introducing the original constraints. For general con-straints, however, appropriate penalty functions are not known in advance and need tobe “discovered.” A simple procedure (see for instance Hammer and Rudeanu 1968;Hansen 1979; Hansen et al. 1993; Boros and Hammer 2002) for finding an appropriatepenalty for any linear constraint is given as follows:

Consider the general constrained problem of the form

min x0 = x Qx

s.t. Ax = b, x binary (1)

This model accommodates both quadratic and linear objective functions since the lin-ear case results when Q is a diagonal matrix (observing that x2

j = x j when x j is a0-1 variable). Under the assumption that A and b have integer components, problemswith inequality constraints can also be put in this form by representing their boundedslack variables by a binary expansion. These constrained quadratic optimization mod-els are converted into equivalent UQP models by adding a quadratic infeasibilitypenalty function to the objective function in place of explicitly imposing the constraintsAx = b.

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Specifically, for a positive scalar P:

x0 = x Qx + P (Ax − b)t (Ax − b)

= x Qx + x Dx + c

= x Q̂x + c

where the matrix D and the additive constant c result directly from the matrix multipli-cation indicated. Dropping the additive constant, the equivalent unconstrained versionof the constrained problem becomes

U B Q P : min x Q̂x, x binary (2)

A suitable choice of the penalty scalar P can always be chosen so that the optimalsolution to UBQP is the optimal solution to the original constrained problem. Forease of reference, the preceding procedure that transforms (1) into (2) will be calledTransformation #1.

Transformation #1 can be used in cases where an appropriate quadratic penaltyfunction isn’t known in advance. In certain special cases, as mentioned earlier, appro-priate penalties are known and can be directly employed. One particularly importantcase that arises in many constrained combinatorial problems is:

x j + xk ≤ 1 (3)

denoting a situation where a pair of binary choices are available and we must precludechoosing both. As shown in the preceding table, an equivalent quadratic penalty forthis situation is simply

Px j xk (4)

Due to the frequency with which the constraint of (3) appears in many importantapplications we single it out for special attention and refer to the penalty of (4) as analternative to the constraint of (3) as Transformation #2. Many of the applications thatfollow were originally modeled as constrained 0/1 models and were recast into theform of UBQP by using Transformation 1 and/or 2.

Other paths to reformulation exist as well. Often a well-chosen change of variablecan result in transforming a constrained model into the form of UBQP. This is par-ticularly important in the context of certain optimization problems on graphs wherebinary variables denoting whether or not an edge is chosen can be replaced by theproduct of the two associated binary node variables. In making such a substitution, wego from an “edge-oriented” model to a “node-oriented” model. This typically resultsin a much smaller model in terms of both the number of variables and the number ofconstraints.

The clique partitioning problem affords a good example for illustrating thisapproach. The standard integer programming (IP) formulation (see for example Oostenet al. 2001) for clique partitioning is:

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max∑

(i, j)∈E

wi j xi j

s.t.

xi j + xir − x jr ≤ 1 ∀ all distinct i, j, r ∈ V

xi j ∈ {0, 1} f or all {i, j} ∈ E

The variable xi j is equal to 1 if the edge (i,j) is in the partition and is equal to 0otherwise. The coefficient wi j is the weight of the edge (i,j) in the graph.

An alternative model results by changing from edge-based variables to node-basedvariables. For this new model we add artificial edges as needed to produce a completegraph and denote an upper bound on the number of cliques to be formed by Kmax.Then, letting xik = 1 if node i is assigned to clique k and xik = 0 otherwise, anequivalent model is:

maxn−1∑

i=1

n∑

j=i+1

wi j

K max∑

k=1

xik x jk

s.t.k max∑

k=1

xik = 1 f or i = 1, n

In this formulation n is the number of nodes in the graph and wi j again denotes theweight of edge (i,j). This model is much smaller than the standard IP in terms of bothnumber of variables and the number of constraints. Note also that it is of the form:

max x ′Qx

s.t.

Ax = b x binary

which can be re-cast into the form of UBQP using Transformation #1.

2.3 Specific application instances

Each of the applications presented below were originally modeled as a constrainedcombinatorial problem and then re-cast into the form of UBQP. Once in this unifiedform the problems were successfully solved by various heuristic means.

Lewis et al. (2005) address the problem of assigning tasks to processors in a dis-tributed, multitasking computer architecture such that the sum of the resultant taskcompletion costs and inter-task communication costs are minimized. The standardmodel for this problem is a constrained quadratic optimization model in binary vari-ables with constraints ensuring that each task gets assigned to one of the processors

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available. The authors employ Transformation #1 to re-cast this model into the formof UBQP which in turn is solved with a basic tabu search heuristic. Computationalexperience with large-scale instances highlights the attractiveness of this approach.

Kochenberger et al. (2005b) discuss the classic vertex coloring problem and how itcan be effectively modeled and solved in the form of UBQP. A standard representa-tion of the K-Coloring problem consists of two categories of constraints, one ensuringthat each node gets a color, and the other ensuring that adjacent nodes receive dif-ferent colors. The authors use Transformation #1 on the first set of constraints andTransformation #2 on the second to produce a UBQP representation of the problem.Computational experience applying a tabu search method to standard test problemsfrom the literature indicates that this approach is very competitive with, and oftensuperior to, specialized methods for vertex coloring.

Similar reformulations, using transformations #1 and/or #2 have been reported forother well-known combinatorial problems. Kochenberger et al. (2005c) examine theuse of UBQP as a tool for clustering microarray data into groups with high degreesof similarity. Wang et al. (2006) discuss the problem of grouping machines and partstogether in a flexible manufacturing system in a manner that facilitates economiesin time and cost. Kochenberger et al. (2007) discuss the use of UBQP as a tool formodeling and solving the generalized independent set (GIS) problem. In each case theoriginal model was re-cast into the form of UBQP and successfully solved in this newform.

In addition, Lewis et al. (2008) discuss the classic set partitioning (SP) problem andhow the UBQP framework can be utilized for modeling and solving this important classof problems. Computational experience using a basic tabu search heuristic on problemswith up to 15,000 variables and 5,000 rows and various densities is presented withcomparisons drawn with CPLEX. Also in 2008, Alidaee et al. (2008) discuss the useof the UBQP model for representing and solving the well-known set packing problem.Favorable computational experience with a wide variety of set packing problems withup to 2,000 variables and 10,000 constraints is reported.

Lewis et al. (2009) discuss the Linear Ordering (LO) problem and how it can bemodeled and solved as an instance of UBQP. The standard model in the literaturefor LO is a large 0-1 linear program with many constraints. For instance, a modeldesigned to order 150 items would have, in the classic model, more than one millionconstraints. Rather than use the general procedure of Transformation #1, the authorsshow how to easily re-cast the constrained model into the unconstrained form of UBQPby using a special quadratic penalty that is uniquely suitable for the problem at hand.Computational experience with both medium and large sized test problems reveals theeffectiveness of this approach.

Douiri and Elbernouss (2012) discuss the Sum Coloring Problem which generalizesthe classical vertex coloring problem by seeking a valid coloring of vertices such thatthe sum of the colors assigned to all vertices in minimized. The transformation toUBQP is accomplished by using Transformation #1 on the constraints that ensureeach vertex gets a color and Transformation #2 on the constraints that require adjacentnodes to have different colors. Computational experience with the resulting UBQPmodel was carried out using a genetic algorithm. Results obtained on a variety ofstandard test problems illustrate the attractiveness of this approach.

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Wang and Xu (2013) discuss another variant of the classical vertex coloring prob-lem, called the Robust Graph Coloring Problem (RGCP), where for a given feasiblecoloring, a penalty is incurred for each non-adjacent vertices that have the same colorassigned. The optimization problem is to determine a feasible coloring that minimizesthe sum of the penalties associated with the edges in the complementary edge set withendpoints that are assigned the same color. As with the previously discussed coloringproblems, the transformation to UBQP is carried out using a combination of Transfor-mation #1 and Transformation #2. Computational experience with several variationsof a genetic algorithm on a set of test problems illustrates the effectiveness of theUBQP approach for modeling and solving RGCP.

Lewis et al. (2013) discuss the use of UBQP for modeling and solving the Gen-eralized Vertex Covering Problem (GVCP). GVCP generalizes the minimum weightvertex covering problem by employing a three tier cost structure for each edge andcharging a cost depending on whether one, both or neither end point of a given edge iscovered by the subset chosen. The optimization problem is to find the subset of nodesthat minimizes the sum of both the node and edge costs. The model presented previ-ously in the literature for GVCP is a large 0-1 linear program with a binary variablefor each node in the graph and two binary variables for each edge. The authors hereshow how GVCP can readily be formulated as UBQP by employing a simple changeof variable such that all edge variables and all constraints are eliminated. Computa-tional experience comparing the original linear model and the equivalent UBQP modelillustrates the superiority of UBQP for this class of problems.

3 Solution methods

While a few special cases of UBQP are polynomially solvable (see for instance Picard1976; Barahona 1986; Pardalos and Jha 1991), UBQP in general is an NP-hard problem(see Pardalos and Jha 1992) and for all but small to moderate sized problems, heuristicmethods are required to produce good solutions in a reasonable amount of computertime. Nonetheless, there is a sizable literature on exact methods for UBQP. In thesections below we first survey the exact methods that have appeared in the literaturefollowed by the heuristic methods described in the literature for solving UBQP.

3.1 Exact methods

The literature on exact methods for UBQP introduces a variety of algorithms, eachwith the virtue of terminating, given enough time and memory, with a globally optimalsolution. Most approaches involve a tree search of a general branch-and-bound naturebut other methods exist as well. In this section we survey, in chronological order, theprominent methods reported over the past thirty–plus years.

Gulati et al. (1984) describe a branch and prune algorithm for solving UBQP whichis designed to determine all local minimizing points, terminating with the globaloptimal solution revealed as the incumbent. Computational experience with randomtest problems with up to 125 variables is given.

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Carter (1984) proposes a branch and bound algorithm for UBQP that first employsmodified form of Cholesky factorization to transform an indefinite instance of UBQPinto an equivalent positive definite form of the problem. Variable elimination basedon hessian information is used accelerate search process. Computational experienceon a variety of random test problems with various characteristics and with up to 30variables is given.

Williams (1985) describes a branch and bound algorithm for UBQP that success-fully solved a set of randomly generated test problems with up to 100 variables. Thealgorithm begins with a reduction procedure that obtains a good starting solution andsubsequently uses the “roof dual” to help guide the depth-first branch and boundsearch. Comparisons with other methods are given.

Barahona et al. (1989) describe an approach to solving UBQP that first reducesthe problem to an equivalent instance of a max-cut problem. This, in turn, is solvedby a linear programming-based branch and bound method. Constraints based on thecut polytope are used to improve node information and enhance the search process.Computational experience is reported on random problems up to size 100 variablesalong with comparisons with other methods.

Kalantari and Bagchi (1990) describe the adaptation of an algorithm for mini-mizing linearly constrained concave quadratic functions for the purpose of solvingUBQP. Their method starts with a transformation to ensure the Q matrix is positivedefinite, giving an equivalent concave quadratic minimization problem. The authorsthen describe their branch-and-bound method where subproblems are defined by fix-ing a variable at zero or one and bounds are computed by minimizing a linear convexenvelope over the feasible region of the subproblem. Computational testing is reportedon random problems with up to 50 variables.

Pardalos and Rodgers (1990a) describe a branch-and-bound algorithm for solvingUBQP that successfully solved a variety of test problems with up to 200 variables.The algorithm, which uses no multiplications or divisions, incorporates dynamic pre-processing techniques for fixing variables and heuristics for finding good startingpoints. In (1990b) the authors describe a parallel version of the algorithm imple-mented and tested on a hypercube architecture. Computational experience and ananalysis of the speedup achieved are presented. Then in (1992), the authors describe avariation of their branch-and-bound algorithm designed for the UBQP representationof the maximum clique problem. Extensive computational experience is reported foralternative branching rules and data structures, in route to producing a specializedalgorithm optimized for the maximum clique problem with up to 1,000 vertices and150,000 edges.

Billionnet and Sutter (1994) describe a branch-and-bound algorithm for solvingUBQP that successfully solved a large variety of random test problems with up to100 variables. Their innovation was in the calculation of lower bounds to guide thesearch process. At each node in the search tree, a lower bound is computed by combin-ing information obtained from roof duality, a quadratic posiform associated with thedirected cycles of an implications graph, and a component obtained from the inducedposiform of degree 4. Computational experience is given comparing the method withother methods.

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Palubeckis (1995) describes a branch-and-bound algorithm for solving UBQP uti-lizing heuristically generated subproblem solutions that are mapped onto the zeron-vector leading to transformed subproblems in the form of the original UBQP model.Special classes of polytope facets are employed in computing bounds used to guidethe search process. Computational experience with random problems with up to 100variables, and additional experience with several real problems having to do withprinted circuit board design, illustrate the efficiency of the method and indicate that itcompares favorably with other contemporary methods.

Helmberg and Rendl (1998) describe a branch-and-bound algorithm for solvingUBQP based on semidefinite relaxations and cutting planes to enhance the quality ofthe bounds produced. The semidefinite relaxations are solved by an interior point algo-rithm specialized for semi-definite programs. Computational experience is reportedon a set of UBQP instances of the max-cut variety with up to 100 variables. While theapproach was robust in that it was successful in solving the test problems attempted,run times were generally not competitive with other recently reported exact methodsfor UBQP.

Hansen et al. (2000) describe an enhanced version of the branch-and-bound methodof Pardalos and Rodgers (1990a) that led to favorable comparisons with the originalalgorithm on a standard set of problems with up to 100 variables. The new methodemploys improved bounds obtained by first transforming the problem to an equivalentposiform which yields tighter roof dual bounds as well as effective variable eliminationtest that efficiently guide the tree search process. The roof dual bounds are computedvia a maximum flow algorithm.

Huang et al. (2006) describe a depth-first branch-and-bound method that begins byfirst formulating an equivalent bi-level formulation of UBQP. This new formulationfacilitates bounding procedures and pruning strategies, utilizing a gradient midpointmethod that proved to be effective in early testing. Computational experience withrandom test problems of various densities and with up to 60 variables is presented.

Pardalos et al. (2006) discuss the connections between discrete optimization andcontinuous optimization in general with a focus on formulations that embed the initialdiscrete domain into a larger continuous space. The authors then focus on the gen-eral UBQP model, indicating how a reformulation based on an appropriate diagonalperturbation, causing the Q matrix to be negative semidefinite, yields an equivalentcontinuous problem of minimizing a quadratic concave function over the unit hyper-cube. A discussion of this approach applied to the maximum clique problem is given.

Pan et al. (2008) describe a continuous approach for solving UBQP based on theFischer-Burmeister nonlinear complementarity function. Rather than employing relax-ations and bounding information in a tree search scheme, the authors reformulateUBQP as a continuous problem with equilibrium constraints. In turn, the optimalsolution to this model is found by a global continuation algorithm utilizing a strictlyconvex global smoothing function and solving a sequence of unconstrained minimiza-tion problems. Computational experience is reported with random problems with upto 1,000 variables, indicating the effectiveness of this approach.

Gueye and Michelon (2009) present a general framework for constructing lineariza-tions of UBQP which, in turn, can be solved in principle by standard optimizers formixed integer linear programs. The framework, which contains existing lineariza-

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tion methods in the literature as special cases, consists of decomposing the objectivefunction into component matrices, identifying a complete linear representation of thepolytope for each component, and then adding constraints that link the componentstogether. A new linearization, derived from the general framework, is described andcomputational comparisons are given with existing methods illustrating the potentialof the new approach.

Pham Dinh et al. (2010) discuss a new continuous approach for solving UBQPthat is based on DC (difference of convex functions) programming. In their approach,principles of DC programming are used to develop a local optimization algorithm(DCA) that solves a finite number of linear programs leading to a locally optimalsolution. Globally optimal solutions are produced by embedding DCA in a branch-and-bound algorithm. Computational experience on problems from the literature withup to 100 variables, along with comparison with other methods, is given.

Mauri and Lorena (2011) present a new algorithm for solving UBQP based onLargangean decompositions. Their method starts with a linearization of UBQP rep-resented by a graph. In turn, the graph is partitioned into clusters of vertices forminga dual problem that is solved by a subgradient algorithm. Clusters are formed usingthe well-known METIS heuristic and the linear Lagrangean subproblems are solvedusing CPLEX. In (2012a) the authors present a column generation alternative to thesubgradient algorithm leading to performance improvement. In (2012b) the authorspresent and test further enhancements to their column generations approach for solvingUBQP. Throughout all, computational experience on standard UBQP test problemswith up 500 variables is given. Comparisons with other decomposition-based methodsare given indicating the potential of the procedures proposed here.

Li et al. (2012) present a new algorithm for solving UBQP based on the inherentgeometric properties of the minimum circumscribed sphere containing the ellipsoidalcontour of the objective function. Based on these properties, effective bounding infor-mation as well as new procedures for optimally fixing variables are derived. In addition,this geometric approach led to new optimality conditions for UBQP. The new bound-ing techniques and variable fixing conditions were combined in a branch-and-boundmethod and tested on standard problems with up to 200 variables. Comparisons drawnwith other recent methods in the literature indicate the attractiveness of the methodproposed.

In addition to the exact methods surveyed above from the literature, we point out thatseveral commercial methods, based on branch-and-cut techniques, are now availableand hold considerable promise for directly optimizing moderate sized instances ofUBQP. See for example the paper by Billionnet and Elloumi (2007) which reports onthe use of the branch-and cut quadratic integer optimizer available from CPLEX.

3.2 Heuristic and Metaheuristic methods

The NP-hard nature of UBQP along with its application potential has motivated a largenumber of papers in recent years describing various heuristic methods for quicklyfinding high quality solutions to medium to large sized problem instances. Althougha few of these methods are simple enough to qualify as heuristics, those that generate

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the best solutions are metaheuristic procedures that incorporate compound strategiesconsiderably more advanced than in the basic heuristics. These methods are surveyedbelow:

Boros et al. (1989) develop a Devour Digest Tidy-up (DDT) procedure to rapidlyobtain a solution to UBQP. Based on the posiform expression of UBQP’s objectivefunction, the proposed method includes devour, digest and tidy-up phases. The devourphase identifies a term with the largest coefficient and sets it to 0 in terms of minimiza-tion. The digest phase draws logical conclusions for the items from the devour phase.The tidy-up phase finally substitutes the logical consequences previously derived intothe current quadratic function. Computational experience indicates the effectivenessof the method proposed, in particular on problems of low density.

Glover et al. (1998) propose an adaptive memory tabu search algorithm, whichincorporates a strategic oscillation scheme to enable the search to go beyond the localoptimum obtained by constructive and destructive phases. A key feature of this methodlies in the use of a critical event memory, that collects recency and frequency informa-tion from critical events (moves that causes the solution values to decrease), to guidethe oscillation process. Another feature lies in the use of adaptive oscillation depths.Extensive computational experience discloses that the proposed method outperformsthe best exact and heuristic methods previously reported in the literature in terms ofspeed and solution quality.

Glover et al. (1999) describe an enhanced version of their previous adaptive memorytabu search algorithm. A simple but effective scheme is proposed for accelerating theevaluation of moves and for updating associated problem information. In addition,methods for generating high quality initial solutions and for creating additional trialsolutions at critical events are also introduced. Computational experience with up to1,000 variables reveals this enhanced version can produce high quality solutions withinseveral minutes.

Beasley (1998) adapts tabu search and simulated annealing to solving UBQP. Thetabu search implementation incorporates a strategy in which, once an improved solu-tion is found, a simple local search is successively employed to perform moves irre-spective of their tabu status. Contrary to the tabu search procedure, a local search isapplied only at the end of the simulated annealing process. Computational compar-isons indicate that their tabu search generally performs better than simulated annealingfor small and medium instances but worse for large instances.

Alkhamis et al. (1998) present a simulated annealing based heuristic with a well-selected cooling schedule. Tested on several hundred test problems, the proposedheuristic outperforms several algorithms based on bounding techniques, in particularwith respect to computational time. Additional analysis shows that initial solutionsand the matrix density have limited influence on the effectiveness of the simulatedannealing algorithm.

Merz and Freisleben (1999) devise a hybrid genetic algorithm, in which a simplelocal search is incorporated into the traditional genetic algorithm. The crossover oper-ator is a variant of uniform crossover, requiring the generated offspring solutions tohave the same hamming distance from the parents. The population updating criterionrefers to the quality of solutions, assuring that each solution occurs only once in thepopulation, as customarily done in scatter search methods. A diversification compo-

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nent is launched when the average hamming distance of the population drops below athreshold or the population is not updated for more than 30 consecutive generations.Computational experience shows that a simple genetic algorithm is sufficient to findbest known results for problem instances with less than 200 variables but for thosewith a large number of variables, local search is needed for attaining high qualitysolutions.

Amini et al. (1999) present a scatter search approach, which consists of a diversi-fication generation method, a solution improvement method, a reference set updatemethod, a subset generation method and a solution combination method. The diversi-fication generation method systematically generates a collection of diverse trial solu-tions based on a seed solution by setting an incremental parameter that determineswhich bits of the seed solution should be flipped. The improvement method performsa compound move that sequentially cycles among three types of candidate moves untilno attractive move can be identified. The reference set update method replaces solu-tions in the reference set with new candidate solutions using the quality measurement.The solution combination method uses linear combination of solutions in a subsetderived from the subset generation method to produce new solutions. Since somevariables may receive fractional values in the solution obtained in the linear combi-nation, a rounding procedure is employed to recover integer values. Experiments onthree classes of problems show the attractiveness of the proposed method.

Lodi et al. (1999) present an evolutionary method for solving UBQP. The proposedalgorithm is characterized by the following features. First, a preprocessing phase isapplied to fix certain variables at their optimal values and keep them unchanged duringeach successive round of local search, hence resulting in a reduced problem scale. Sec-ond, a local search procedure that alternates between construction phase and destruc-tive phases is used to get an improved solution. Finally, a uniform crossover operator isused to produce offspring solutions, where variables with common values in parentalsolutions are temporarily fixed in this round of local search. Computational experienceon problem instances with up to 500 variables is given. A further analysis demonstratesthat the preprocessing phase is effective for small problem instances but is unable toappreciably reduce the problem size for large ones.

Katayama et al. (2000) propose a genetic local search algorithm for solving UBQP.Their local search procedure integrates 1-flip moves dedicated to going into new goodsearch area and k-flip moves dedicated to solution improvement. A traditional uniformcrossover and a simple mutation operator are joined to generate a suitable offspringsolution. A diversification/restart strategy is incorporated to maintain a diversifiedpopulation. Tests on large problem instances indicate the effectiveness of the proposedalgorithm.

Katayama and Narihisa (2001) present a simulated annealing algorithm with aninnovative use of multiple annealing processes to enhance the search. Each annealingprocess takes the best solution found in the previous annealing process as the initialsolution and employs a different initial temperature. Experimental results demonstratethe performance of the proposed algorithm, especially for large instances with 2,500variables.

Merz and Freisleben (2002) describe a greedy heuristic and two local search algo-rithms based upon 1-flip and k-flip neighborhoods. The greedy construction procedure

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starts from a solution with all variables assigned to 0.5 (the so called third state) andeach constructive step picks a variable with probability proportional to the gain valuewhen changing the variable’s value from 0.5 to 0 or 1. Each iteration of the 1-fliplocal search proceeds to the neighbor solution with the best solution quality. The k-flip local search borrows the idea from the Lin-Kernighan algorithm of Kernighanand Lin (1970) for solving the graph partitioning problem to efficiently reduce theneighborhood exploration. Each k-flip move consists in repeating performing the best1-flip move until all 1-flip moves are performed and picking the best from the resultingsolutions. Computational comparisons disclose the superiority of the multistart k-fliplocal search combined with randomized greedy initial solutions.

Glover et al. (2002) propose several one-pass heuristics to advance the DevourDigest Tidy-up (DDT) method of Boros et al. (1989). Based on the hypothesis thatsetting multiple variables with value 1 or 0 in a pass may lead to worse performance, theidea is to guarantee only one variable gets the implied assignment in each pass. The pro-posed one-pass heuristics differ in strategies for evaluating contributions of variables.Computational experience indicates that the method outperforms the DDT method butno single one-pass heuristic dominates the others on every problem instance.

Palubeckis and Tomkevicius (2002) present a greedy random adaptive search pro-cedure (GRASP) which cycles between a construction phase and a local search phase.Each step in the construction phase picks a variable from a candidate list with probabil-ity proportional to the gain value of this variable, where the candidate list is composedof a certain number of variables with the largest gain values, calculated accordingto a specific gain function. The local search phase implements a simple ascent algo-rithm. Two enhanced versions are tested, which result by replacing local search withtabu search and by combining a classic random restarting procedure with tabu search.Computational comparisons illustrate the merit of incorporating greedy constructionbased initial solutions and tabu search.

Palubeckis (2004) examines five multistart tabu search strategies dedicated to theconstruction of an initial solution. The first multistart strategy produces a new initialsolution in a random way. The second identifies a candidate set of variables whosevalues are prone to change when moving from the current solution to an optimal oneand then applies a steepest ascent algorithm where variables not included in this can-didate set are fixed at specific values. The third multistart strategy is the same as theconstructive phase proposed in Palubeckis and Tomkevicius (2002). The fourth usesa set of elite solutions to calculate the probability of each variable being assignedvalue 1. If the probability for a given variable is larger than 0.5, then this variableis assigned to be 1 in the constructed solution; otherwise it is assigned to be 0. Thelast multistart strategy uses a perturbation scheme of changing the problem instanceat hand, followed by a short run of tabu search on the modified instance. Extensivecomparisons on problem instances with up to 7,000 variables demonstrate the algo-rithm using the second multistart strategy performs better than the other proposedalternatives.

Merz and Katayama (2004) conduct landscape analysis and observe that (1) localoptima of the UBQP problem instances are concentrated in a small fraction of thesearch space; (2) the fitness of local optima and the distance between local optima andthe global optimum are correlated. Based on the observations, they propose a memetic

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algorithm in which an innovative variation operator is used to generate an offspringsolution and the k-flip local search proposed in Katayama et al. (2000) is used toimprove solution quality. The variation operator introduces new alleles not containedin both parents by referring to the move gain of performing 1-flip moves, avoidingthe rediscovery of local optima already extensively visited. Comparisons with otheralgorithms demonstrate the effectiveness of the proposed algorithm.

Boros et al. (2006) present several preprocessing techniques to simplify the UBQPproblem. The purpose of the preprocessing simplification is to provide several fea-tures, including lower bounds for the minimum of the objective function, optimalassignments for some variables, and binary relations between the values of certainpairs of variables and subproblems decomposed from the original problem. Thesimplification is achieved by using basic techniques such as first-order derivatives,second-order derivatives or roof-duality, and by using integrative techniques thatcombine the conclusions derived from the basic techniques. Computational experi-ence on numerous problem classes shows the value of the proposed preprocessingtechniques.

Palubeckis (2006) presents an iterated tabu search algorithm which uses a dedi-cated perturbation mechanism to enhance the high-quality solution obtained by thetabu search procedure. Each step of the perturbation constructs a candidate list of alimited size consisting of variables with largest 1-flip move gains with regard to thiscurrent solution, from which a variable is randomly selected and flipped to complementthe value of this variable. The current solution is thus updated and the next pertur-bation step continues until the number of perturbed variables reaches the specifiednumber. Comparisons with state-of-the-art algorithms disclose the competitiveness ofthe proposed algorithm in spite of its simplicity.

Boros et al. (2007) present a local search scheme for solving UBQP. Starting froman initial solution, each iterative step constructs a candidate set from which a variableis picked and its value is changed to its complement, thus moving to the next solu-tion. This iterative procedure repeats until the candidate set becomes empty. Basedon the above scheme, they investigate five initialization methods, two candidate setconstruction methods and four variable selection methods, thus reaching up to 40local search alternatives. Experiments on multiple benchmark instances indicate thatthe local search alternative combining the following methods achieves the best perfor-mance. The initial method assigns each variable with a fractional value equaling to theproportion of the sum of all the positive entries of the matrix in the sum of the absolutevalue of each entry of the matrix. The candidate set construction method constructs acandidate set consisting of variables that yield an improvement in the current solutionby flipping its value regardless of whether or not it was already flipped in the previousiteration. The variable selection method selects from the candidate set the variablewith the largest improvement to the current solution.

Glover et al. (2010) present a diversification-driven tabu search algorithm, whichalternates between a basic tabu search procedure and a memory-based perturbationstrategy guided by a long-term memory. Three memory structures are introduced inthe perturbation strategy: (1) a flipping frequency vector to record the number oftimes a variable has been flipped from the initial iteration until the current iteration;(2) an elite set of solutions to record a certain number of best local optimal solutions;

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(3) a consistency vector to count the times each variable is assigned a given value inthe set of elite solutions. Based on the memory information, the perturbation opera-tor modifies an elite solution by favoring variables with low flipping frequency andhigh consistency to flip. Comparisons drawn with several algorithms proposed byPalubeckis (2004, 2006) disclose the superiority of this algorithm.

Lü et al. (2010a) present a hybrid metaheuristic approach which has the followingfeatures. First, it combines a traditional uniform crossover operator with a diversi-fication guided path relinking operator to guarantee the quality and diversity of anoffspring solution. Second, it defines a new distance by reference to variable’s impor-tance and employs a quality-and-distance criterion to update the population. Finally, atabu search procedure is responsible for intensified examination around the offspringsolutions. Computational comparisons with best performing algorithms indicate theeffectiveness of this hybrid algorithm.

Lü et al. (2010b) develop a hybrid genetic tabu search with multi-parent crossoverto solve UBQP. The proposed algorithm jointly uses traditional uniform crossoverand logic multi-parent combination operators to generate diversified offspring solu-tions. Computational experience is given showing the competitiveness of the proposedalgorithm.

Cai et al. (2011) present a memetic clonal selection algorithm with estimationof distribution algorithm (EDA) guided vaccination for solving UBQP. The proposedalgorithm adopts EDA vaccination, fitness uniform selection scheme and adaptive tabusearch to overcome the deficiencies of traditional clonal selection algorithm. Experi-mental comparisons indicate the tabu search algorithm enhances the performance ofthe clonal selection algorithm.

Shylo and Shylo (2011) develop a global equilibrium search which performs multi-ple temperature cycles. Each temperature cycle includes an initial solution generationphase and a tabu search phase. The method to generate an initial solution employshistorical information to determine the probability that a variable receives the value1. The tabu search procedure requires that each admissible move leads to a solutionwith hamming distance to a reference set surpassing a distance threshold. Computa-tional comparisons with several algorithms indicate the attractiveness of the proposedalgorithm.

Hanafi et al. (2013) devise five alternative DDT heuristics based on different rep-resentations of the BQO formulation. DDT1 to DDT4 respectively have standard,posiform, bi-form and negaform representations and DDT5 has a posiform repre-sentation combined with a one-pass mechanism. One obvious difference betweenthe DDT alternatives proposed here and those proposed by Boros et al. (1989) andGlover et al. (2002) lies in the use of an r-flip local search procedure to improvesolutions obtained by DDT constructions. Extensive tests on small, medium and largebenchmark instances disclose that (1) DDT3 with the bi-form representation generallyproduces the best results for medium and large instances; (2) the r-flip local searchcontributes to significant improvement of the results of the proposed DDT methodswith only a slight increase of time consumption.

Wang et al. (2011) present a tabu Hopfield neural network with an estima-tion of distribution algorithm (EDA). The cooperation between long term mem-ory of EDA with the short term memory of tabu search prevents the network from

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becoming trapped in local optima. Computational testing indicates the superior-ity of the proposed algorithm compared to other Hopfield neural network basedalgorithms.

Lü et al. (2011) study neighborhood union and token-ring search methods withina tabu search algorithm. They focus on two neighborhoods, N1 consisting of 1-flipmoves and N2 consisting of a chosen subset of 2-flip moves. The neighborhood unionincludes the strong neighborhood union that picks each move from both N1 and N2and the selective neighborhood union that picks a move from N1 with probability pand N2 with probability 1 − p. The token ring search continuously performs movein N1 until no improvement is possible and then switches to perform move in N2 tocontinue the search. Computational comparisons reveal the superiority of the tokenring search over the neighborhood union.

Wang et al. (2012a,b,c) present two path relinking algorithms, which are composedof a reference set construction method, a tabu search based improvement method, areference set update method, a relinking method and a path solution selection method.The proposed algorithms differ from each other mainly on the way they generate thepath, one employing a greedy strategy and the other employing a random strategy.Extensive computational experience and comparisons with several state-of-the-artalgorithms highlight the attractiveness of the proposed algorithms in terms of bothsolution quality and computational efficiency.

Wang et al. (2012a,b,c) propose a simple GRASP-Tabu Search algorithm workingwith a single solution and an enhanced version by combining GRASP-Tabu Searchalgorithm with a population management strategy based on an elite reference set.In the basic version, the initial solution is constructed according to a greedy randomconstruction heuristic. In the enhanced version, a new solution is reconstructed by firstinheriting parts of the good assignments of one elite solution to form a partial solutionand then completing the remaining parts as the basic version does. Experimentaltests on a large range of both random and structured problem instances disclose thatthe proposed algorithms, in particular the enhanced version, yield very competitiveoutcomes.

Wang et al. (2012a,b,c) present a backbone guided tabu search algorithm whichalternates between a basic tabu search procedure and a variable fixing/freeing phasebased on identifying strongly determined variables. While the tabu search phaseensures the exploitation of a search space, the variable fixing (freeing) phase dynami-cally enlarges (reduces) the backbone of assigned values that launches the tabu searchexploration. Experiments show that the proposed algorithm obtains highly competi-tive outcomes in comparison with the previous best known results from the literature.A direct comparison with the underlying tabu search procedure confirms the merit ofincorporating backbone information.

As indicated in the papers of this section, our ability to efficiently solve largeinstances of UBQP by heuristic means has grown substantially in recent years. Itis common now for authors to report computational experience on problems with7,000–10,000 variables. Note that the set partitioning application discussed in Sect.2.3 by Lewis et al. (2008) reported computational experience on problems up to 15,000variables.

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4 Key theoretical results

By far the majority of the papers in the literature related to UBQP are primarily devotedto applications or various solution schemes, either exact or heuristic in nature. As aresult, our priority in this paper has been to focus our survey on applications andsolution methodologies. Many of the articles surveyed in Sect. 3 above, however,contain a discussion of the theoretical results relevant to the method being put forth.That is, these papers are mainly about the method at hand but may also contain adiscussion of underlying theory. As a result, we’ve not explicitly highlighted theoreticalissues but rather left them to be discovered, as might be appropriate, as part of thearticles on applications and solutions methods surveyed. Nonetheless, there are a fewrecent papers in the literature of particular note focused on theoretical issues pertainingto UBQP. It is these papers that we highlight here in this section.

Carraesi et al. (1999) present an exact algorithm for testing the optimality of a givensolution for a quadratic 0-1 unconstrained problem. Their method, based on necessaryand sufficient conditions introduced by Hirriart-Urruty for general convex problems,expands their earlier work (1995) which was an approximation scheme for testingsolutions.

Beck and Teboulle (2000) characterize global optimal solutions for UBQP as wellas discussing the relationship between optimal solutions to UBQP and the optimalsolutions of its continuous relaxation. They derive a sufficient optimality conditionwhich guarantees that a given feasible point is a global optimal for UBQP as well asa necessary global optimality condition.

Jeyakumar et al. (2007) examine the relationship between the global optimality ofnonconvex constrained optimization and Lagrange multiplier conditions, establishingsufficient as well as necessary conditions for global optimality for general quadraticminimization problems with quadratic constraints. This analysis led, as a special case,to new sufficient and necessary global optimality conditions for UBQP that are sharperthan those given earlier by Beck and Teboulle.

Xia (2009), by analyzing local sufficient optimality conditions, also extended theBeck and Teboulle results by developing tighter sufficient optimality conditions. Inaddition, without making the positive-semidefinite assumption, Xia examines the rela-tionship between local/global minimizers of UBQP and the KKT points of the con-tinuous relaxation, further extending previous results in the literature.

Gao and Ruan (2010) present a discussion of canonical duality theory, designedin general for a wide class of nonconvex/nonsmooth/discrete problems. The authorsshow how this duality theory can be adapted for the quadratic case with binary con-straints. Conditions are given that allow instances of UBQP to be converted intosmooth concave maximization dual problems over a closed convex feasible regionwithout a duality gap. Finally, the relationship between canonical duality theory andsemi-definite programming for UBQP is discussed.

Zheng et al. (2012) present new sufficient conditions for verifying zero duality gapin nonconvex constrained quadratic programs and then show how the results specializefor UBQP. In related work, Sun et al. (2012) investigate the duality gap between UBQPand its semi-definite programming relaxation. Making the connection between the

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duality gap and the cell enumerations of hyperplane arrangement in discrete geometry,estimates of the duality gap can be derived, yielding improved lower bounds for UBQP.

We note that there are several theoretical papers in the literature on the constrainedversion of UBQP that don’t explicitly consider the pure UBQP model but are nonethe-less relevant to our work in that UBQP is a special case of the constrained casesconsidered. Notably, Pinar (2004) gives a discussion of sufficient global optimalityconditions for the problem of minimizing a quadratic function in binary variablessubject to equality quadratic constraints. Lu et al. (2011) presents a discussion ofKKT conditions and conic relaxations to develop sufficient conditions that generalizeknown positive semi-definiteness results for finding globally optimal solutions for theproblem of minimizing a UBQP subject to inequality quadratic constraints. Finally,Li (2012) presents an extension of Pinar’s global optimality conditions for the quadraticequality constrained case along with presenting conditions enabling global optimalityto be assessed by checking the positive semi-definiteness of a related matrix.

5 Summary and conclusions

Interest in UBQP has grown substantially in recent years as researchers have discov-ered the remarkable ability of this simple model form to represent a wide variety ofcombinatorial problems along with its computational challenge, particularly as modelsizes have increased. Due to its NP-hard nature, methods capable of producing exactsolutions are limited to modest sized applications, giving way to modern heuristicmethods for larger models. Even today, exact methods appear to be limited to a fewhundred variables. In an effort to realize the application potential of UBQP as modelsize scales to higher levels, most research is focused on metaheuristic methods of onekind or another. The results are encouraging: Articles in the 80s were reporting onsolving problems with 100–200 variables while more recent articles are reporting onproblems with up to 15,000 variables. To a large extent, this growth in performance isdue to advances in both algorithm design and computer hardware.

Successfully moving to the next order of magnitude in terms of model size willrequire creative schemes for handling very large Q matrices along with improvedalgorithmic methods. Various partitioning and multi-level methods hold particularpromise here but the door is open for other innovations as well. Advances in computerperformance, both in terms of storage and speed, can also be expected to lend ahand in allowing larger applications. Moreover, developments in the area of quantumcomputing, as illustrated by the work by Neven et al. (2008), represent emergingtechnologies with a potential for solving combinatorial problems as represented byUBQP. Future papers will reveal which of these research areas, or indeed, if someother approach, will contribute to facilitating solutions to UBQP as application sizecontinues to scale upward.

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